Compare linear and non-linear representations

Ontario Grade 9 Mathematics

Algebra

Comparing Linear and Non-Linear Representations (C4.1)

Have you ever saved money at the same amount each week, or watched a plant double in height every few days? Both situations show how two quantities can be related — but they behave very differently. In this lesson you will learn to tell apart two types of relationships: linear and non-linear. You will use tables, graphs, and equations to spot the difference, and you will practise explaining your reasoning in clear mathematical language. No new prerequisites are needed beyond what you already know about plotting points and substituting values into simple expressions.

What you will learn

Identify whether a relationship is linear or non-linear using a table of values, a graph, or an equation.
Explain what makes a relationship linear: a constant rate of change between any two points.
Explain what makes a relationship non-linear: a rate of change that is not constant.
Compare a linear and a non-linear representation side by side and justify which is which.

What Is a Relationship Between Two Quantities?

A relationship describes how one quantity changes as another quantity changes. We usually call the two quantities the input and the output. The input is the value you choose or control — for example, the number of weeks. The output is the value that results — for example, the total amount of money saved.

A table of values is one of the simplest ways to record a relationship. You list several input values in one column and the matching output values in another column. A graph shows the same information as a collection of points plotted on a coordinate grid. An equation uses symbols to describe the rule that connects every input to its output.

Before comparing linear and non-linear relationships, make sure you are comfortable with two prerequisite ideas from Grade 8. First, plotting a point

(x, y)

means moving

x

units along the horizontal axis and

y

units along the vertical axis. Second, substituting a value means replacing a letter in an expression with a number and then evaluating — for example, if the rule is

y = 3x + 1

and

x = 4

, then

y = 3(4) + 1 = 13

.

The input is the quantity you choose; the output is the quantity that results.
Tables, graphs, and equations are three representations of the same relationship.
Substitution and point-plotting from Grade 8 are the only prerequisites needed here.

Linear Relationships: A Constant Rate of Change

A relationship is called linear when the output changes by the same amount every time the input increases by one unit. This steady, predictable change is called a constant rate of change. Think of a person walking at a steady speed: for every extra minute they walk, they cover the same additional distance. The output goes up by the same amount each step — no more, no less.

You can spot a constant rate of change in a table by looking at the differences between consecutive output values. Subtract each output from the one below it. If every difference is identical, the rate of change is constant and the relationship is linear. In a table where the input increases by 1 each row, this difference is called the first difference.

On a graph, a linear relationship produces points that lie exactly on a straight line. That straight-line appearance is where the word 'linear' comes from — it comes from the Latin word for line. The equation of a linear relationship always has the form

y = mx + b

, where

m

is the rate of change (sometimes called the slope) and

b

is the starting value when the input is zero. Neither

m

nor

b

can be multiplied by

x

more than once, and

x

cannot appear as an exponent.

y = mx + b

A linear relationship has a constant rate of change.
In a table with inputs increasing by 1, all first differences (output gaps) are equal.
On a graph, the points fall on a straight line.
The equation has the form $y = mx + b$ where $x$ appears to the first power only.

Non-Linear Relationships: A Changing Rate of Change

A relationship is called non-linear when the rate of change is not constant. The output does not go up (or down) by the same amount each time the input increases by one unit. Think of a ball dropped from a height: it falls slowly at first and then much faster. The extra distance covered each second keeps growing — the change itself is changing.

In a table, you can detect a non-linear relationship by checking the first differences. If those differences are not all equal, the rate of change is not constant and the relationship is non-linear. The differences might grow larger, grow smaller, or follow some other uneven pattern.

On a graph, a non-linear relationship produces points that do not all lie on a single straight line. When you connect them in order, the result is a curve — it could bend upward, bend downward, or take some other curved shape. One very common non-linear relationship at this grade level is one where the input is squared, giving an equation like

y = x^2

. Because

x

appears with an exponent of 2, the equation is not in the

y = mx + b

form, and the graph is a curve, not a line.

y = x^2

A non-linear relationship has a rate of change that is not constant.
In a table, the first differences are not all equal.
On a graph, the points do not lie on a straight line — they form a curve.
An equation like $y = x^2$ is non-linear because the exponent on $x$ is greater than 1.

Comparing the Two: A Side-by-Side Look

The clearest way to understand the difference is to look at a linear relationship and a non-linear relationship at the same time, using the same input values. Consider two delivery services. Service A charges a flat connection fee of CAD 2 and then CAD 3 per kilometre. Service B charges based on the square of the distance in kilometres, so the cost equals the number of kilometres squared, plus CAD 1.

Service A follows the rule

y = 3x + 2

, where

x

is kilometres and

y

is the cost. Service B follows the rule

y = x^2 + 1

. Looking at a table for

x

values from 1 to 5, you can calculate each service's cost and then check the first differences to see which one is constant.

For Service A: the costs are CAD 5, CAD 8, CAD 11, CAD 14, CAD 17. The first differences are all

+3

. That constant difference confirms a linear relationship. For Service B: the costs are CAD 2, CAD 5, CAD 10, CAD 17, CAD 26. The first differences are

+3

,

+5

,

+7

,

+9

— they keep growing by 2 each time, so they are not constant. That confirms a non-linear relationship. On a graph, Service A's points form a straight line while Service B's points form a curve that bends upward more and more steeply.

\text{First difference} = y_{\text{next}} - y_{\text{current}}

Checking first differences in a table is a reliable method: equal differences mean linear, unequal differences mean non-linear.
Graphs confirm the finding: a straight line is linear, a curve is non-linear.
The same conclusion can be drawn from the equation: $y = 3x + 2$ is linear; $y = x^2 + 1$ is non-linear.
All three representations — table, graph, equation — must agree with each other.

Reading the Equation to Identify the Type

Sometimes you are given only an equation and asked whether the relationship is linear or non-linear. A straightforward strategy: look at the exponent on the input variable. If the highest exponent on

x

is exactly 1 — meaning

x

appears as just

x

, not

x^2

or any higher power — the relationship is linear. If the exponent on

x

is 2 or higher, the relationship is non-linear.

Examples of linear equations:

y = 5x

,

y = -2x + 7

,

y = \frac{x}{4} - 3

. In every case,

x

is raised only to the power of 1. Examples of non-linear equations at this grade level:

y = x^2

,

y = 3x^2 - 1

,

y = x^2 + 3x

. In each of these,

x

is raised to the power of 2, which is greater than 1, so the relationship is non-linear.

Be careful with equations that look complicated but are still linear. For instance,

y = 4(x - 2) + 6

can be expanded to

y = 4x - 8 + 6 = 4x - 2

. After expanding,

x

still appears only to the power of 1, so the relationship is linear. Expanding before judging is a safe habit.

y = mx + b (linear, highest power of x is 1) vs. y = ax^2 + bx + c (non-linear, highest power of x is 2, where a ≠ 0)

If the highest power of $x$ in the equation is 1, the relationship is linear.
If the highest power of $x$ is 2 or more, the relationship is non-linear.
Expand brackets first, then check the exponent on $x$ .
An equation like $y = 4(x - 2) + 6$ simplifies to $y = 4x - 2$ , which is linear.

Side-by-Side Comparison: Pattern P (Linear) vs. Pattern Q (Non-Linear)

Input $x$	Pattern P output $y = 2x+3$	P First Difference	Pattern Q output $y = x^2-1$	Q First Difference
$0$	$3$	—	$-1$	—
$1$	$5$	$+2$	$0$	$+1$
$2$	$7$	$+2$	$3$	$+3$
$3$	$9$	$+2$	$8$	$+5$
$4$	$11$	$+2$	$15$	$+7$

Worked example

Identifying Linear vs. Non-Linear from a Table and an Equation

Two patterns are described below. Pattern P has the rule

y = 2x + 3

. Pattern Q has the rule

y = x^2 - 1

. (a) Complete a table of values for each pattern using

x = 0, 1, 2, 3, 4

. (b) Calculate the first differences for each pattern. (c) State which pattern is linear and which is non-linear, and give two pieces of evidence for each conclusion.

Step 1 – Calculate the output values for Pattern P
Substitute each input value into $y = 2x + 3$ to find the matching output. Substitution means replacing $x$ with the number and then evaluating the expression.
$x=0: y=2(0)+3=3;\quad x=1: y=2(1)+3=5;\quad x=2: y=2(2)+3=7;\quad x=3: y=2(3)+3=9;\quad x=4: y=2(4)+3=11$
Step 2 – Calculate the output values for Pattern Q
Substitute each input value into $y = x^2 - 1$ . Remember that $x^2$ means $x$ multiplied by itself.
$x=0: y=0^2-1=-1;\quad x=1: y=1^2-1=0;\quad x=2: y=2^2-1=3;\quad x=3: y=3^2-1=8;\quad x=4: y=4^2-1=15$
Step 3 – Find the first differences for Pattern P
Subtract each output from the output directly below it in the table. If all these differences are equal, the rate of change is constant.
$5-3=2,\quad 7-5=2,\quad 9-7=2,\quad 11-9=2$
Step 4 – Find the first differences for Pattern Q
Apply the same subtraction process to Pattern Q's outputs.
$0-(-1)=1,\quad 3-0=3,\quad 8-3=5,\quad 15-8=7$
Step 5 – Check the equation of each pattern
For Pattern P, look at the exponent on $x$ in $y = 2x + 3$ . The exponent is 1, confirming linear. For Pattern Q, the exponent on $x$ in $y = x^2 - 1$ is 2, confirming non-linear.
$y = 2x^{1} + 3 \Rightarrow \text{linear};\qquad y = x^{2} - 1 \Rightarrow \text{non-linear}$
Step 6 – State the conclusion with two pieces of evidence each
Combine the table evidence (first differences) with the equation evidence (exponent on x) to write a complete justification for both patterns.
Pattern P: first differences all = 2 (constant) AND highest power of x = 1 \Rightarrow \textbf{linear} Pattern Q: first differences are 1, 3, 5, 7 (not constant) AND highest power of x = 2 \Rightarrow \textbf{non-linear}

Answer: Pattern P (

y = 2x + 3

) is linear: its first differences are all equal to

2

and the highest power of

x

is

1

. Pattern Q (

y = x^2 - 1

) is non-linear: its first differences are CAD 1, 3, 5, 7 (not constant) and the highest power of

x

is CAD 2.

Check: Verify by re-substituting

x = 4

into each equation. Pattern P:

y = 2(4) + 3 = 11

✓. Pattern Q:

y = 4^2 - 1 = 16 - 1 = 15

✓. Also confirm that

15 - 8 = 7

, matching the last first difference calculated ✓.

Common mistakes and how to avoid them

Checking only one first difference and concluding the relationship is linear if that one difference looks constant.

Correction: Always check every consecutive pair of output values. One equal difference is not enough — all first differences must be equal for the relationship to be linear.

Thinking that a negative rate of change means the relationship is non-linear.

Correction: A constant negative rate of change is still constant, so the relationship is still linear. For example, first differences of

-4, -4, -4, -4

confirm a linear relationship with a negative rate of change.

Deciding a relationship is non-linear just because the equation looks complicated, without expanding brackets first.

Correction: Expand the equation fully before judging. For example,

y = 3(x + 2) - 4

expands to

y = 3x + 6 - 4 = 3x + 2

, which is linear because the highest power of

x

is still

1

.

Confusing the first difference (the gap between outputs) with the output value itself.

Correction: The first difference is calculated by subtracting: next output minus current output. It is not the output value itself. Always subtract consecutive outputs, not the inputs.

Assuming that if the graph curves even slightly it must be non-linear, without verifying using the table or equation.

Correction: Plotting errors or uneven axis scales can make a straight line look curved. Always confirm your conclusion using first differences in a table or the exponent in the equation.

Lesson summary

A linear relationship has a constant rate of change: the output increases (or decreases) by the same amount every time the input increases by one unit.
In a table, equal first differences confirm a linear relationship; unequal first differences confirm a non-linear relationship.
On a graph, a linear relationship produces points on a straight line; a non-linear relationship produces points on a curve.
In an equation, if the highest power of the input variable is $1$ , the relationship is linear (form $y = mx + b$ ). If the highest power is $2$ or greater, the relationship is non-linear.
Always expand brackets in an equation before deciding whether it is linear or non-linear.
All three representations — table, graph, and equation — must give the same conclusion about whether a relationship is linear or non-linear.

Check your understanding

Question 1

A table of values shows outputs of CAD 4, 7, 10, 13, 16 for inputs CAD 1, 2, 3, 4, 5. What are the first differences, and what do they tell you?

The first differences are CAD 3, 3, 3, 3, which means the relationship is linear.
The first differences are CAD 4, 7, 10, 13, which means the relationship is linear.
The first differences are CAD 3, 3, 3, 3, which means the relationship is non-linear.
The first differences are CAD 1, 2, 3, 4, which means the relationship is non-linear.

Show answer and explanation

The first differences are CAD 3, 3, 3, 3, which means the relationship is linear.

Subtract each output from the next:

7 - 4 = 3

,

10 - 7 = 3

,

13 - 10 = 3

,

16 - 13 = 3

. All first differences equal

3

, which is constant. A constant rate of change means the relationship is linear.

Question 2

Which of the following equations represents a non-linear relationship?

$y = 5x - 2$
$y = -3x + 8$
$y = 2(x + 4) - 1$
$y = x^2 + 3x$

Show answer and explanation

y = x^2 + 3x

y = x^2 + 3x

contains

x^2

, so the highest power of

x

is

2

. Any equation where

x

is raised to a power greater than

1

is non-linear. The other three equations all simplify so that

x

appears only to the power of

1

, making them linear.

Question 3

A graph shows two relationships. Relationship A forms a perfectly straight line. Relationship B forms a curve that bends upward. Which statement is correct?

Both A and B are linear because both go upward.
A is linear and B is non-linear, because a straight line indicates a constant rate of change and a curve indicates a changing rate of change.
A is non-linear because straight lines do not curve.
B is linear because it rises faster than A.

Show answer and explanation

A is linear and B is non-linear, because a straight line indicates a constant rate of change and a curve indicates a changing rate of change.

A straight-line graph means the rate of change is constant — that is the definition of a linear relationship. A curve means the rate of change is not constant — that is the definition of a non-linear relationship. The direction (upward or downward) does not determine linearity.

Question 4

A student checks the first differences for a table and finds:

+2, +4, +6, +8

. What conclusion should the student draw?

The relationship is linear because the differences are all positive.
The relationship is linear because the differences are even numbers.
The relationship is non-linear because the first differences are not all equal.
The relationship is non-linear because the differences are greater than 1.

Show answer and explanation

The relationship is non-linear because the first differences are not all equal.

For a relationship to be linear, every first difference must be the same value. Here the differences are CAD 2, 4, 6, 8 — they are not equal; they keep increasing by CAD 2. Because the first differences are not constant, the rate of change is not constant, so the relationship is non-linear.

Key terms

Linear relationship: A relationship where the output changes by the same amount every time the input increases by one unit, producing a straight-line graph and an equation of the form $y = mx + b$ .
Non-linear relationship: A relationship where the rate of change is not constant. The graph is a curve, not a straight line, and the equation has the input variable raised to a power greater than 1.
Rate of change: How much the output changes for each one-unit increase in the input. In a linear relationship this value is constant; in a non-linear relationship it varies.
First difference: The result of subtracting one output value from the next output value in a table where the inputs increase by 1. Equal first differences indicate a linear relationship.
Input: The quantity you choose or control in a relationship, often represented by the variable $x$ .
Output: The quantity that results from applying the rule to the input, often represented by the variable $y$ .
Constant: A value that does not change. In a linear relationship, the rate of change is constant — it stays the same for all input values.
Exponent: A small raised number that tells you how many times a value is multiplied by itself. For example, in $x^2$ the exponent is $2$ , meaning $x \times x$ .

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C4.1. It is a study resource, not an official curriculum publication.

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C4.1 · Compare linear and non-linear representations

Ontario Grade 9 Mathematics

Comparing Linear and Non-Linear Representations (C4.1)

What you will learn

What Is a Relationship Between Two Quantities?

Linear Relationships: A Constant Rate of Change

Non-Linear Relationships: A Changing Rate of Change

Comparing the Two: A Side-by-Side Look

Reading the Equation to Identify the Type

Side-by-Side Comparison: Pattern P (Linear) vs. Pattern Q (Non-Linear)

Identifying Linear vs. Non-Linear from a Table and an Equation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review